Morse–Kelley set theory
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In the
foundations of mathematics Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
that is closely related to
von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collect ...
(NBG). While von Neumann–Bernays–Gödel set theory restricts the
bound variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
s in the schematic formula appearing in the
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
es as well as sets, as first suggested by Quine in 1940 for his system ML. Morse–Kelley set theory is named after mathematicians John L. Kelley and
Anthony Morse Anthony Perry Morse (21 August 1911 – 6 March 1984) was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelle ...
and was first set out by and later in an appendix to Kelley's textbook ''General Topology'' (1955), a graduate level introduction to
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. Kelley said the system in his book was a variant of the systems due to
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem ...
and Morse. Morse's own version appeared later in his book ''A Theory of Sets'' (1965). While von Neumann–Bernays–Gödel set theory is a
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
of
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
(ZFC, the canonical set theory) in the sense that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, Morse–Kelley set theory is a
proper extension In mathematical logic, a conservative extension is a Theory (mathematical logic)#Subtheories and extensions, supertheory of a Theory (mathematical logic), theory which is often convenient for proving theorems, but proves no new theorems about the la ...
of ZFC. Unlike von Neumann–Bernays–Gödel set theory, where the axiom schema of Class Comprehension can be replaced with finitely many of its instances, Morse–Kelley set theory cannot be finitely axiomatized.


MK axioms and ontology

NBG and MK share a common
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
. The
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
consists of classes. Classes that are members of other classes are called sets. A class that is not a set is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
. The primitive
atomic sentence In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences ...
s involve membership or equality. With the exception of Class Comprehension, the following axioms are the same as those for NBG, inessential details aside. The symbolic versions of the axioms employ the following notational devices: * The upper case letters other than ''M'', appearing in Extensionality, Class Comprehension, and Foundation, denote variables ranging over classes. A lower case letter denotes a variable that cannot be a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, because it appears to the left of an ∈. As MK is a one-sorted theory, this notational convention is only
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
. * The monadic
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
\ Mx, whose intended reading is "the class ''x'' is a set", abbreviates \exists W(x \in W). * The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
\varnothing is defined by \forall x (x \not \in \varnothing). * The class ''V'', the
universal class {{other uses, Universe (mathematics) Universal class is a category derived from the philosophy of Hegel, redefined and popularized by Karl Marx. In Marxism it denotes that class of people within a stratified society for which, at a given point in hi ...
having all possible sets as members, is defined by \forall x (Mx \to x \in V). ''V'' is also the
von Neumann universe In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
.
Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
: Classes having the same members are the same class. :\forall X \, \forall Y \, ( \forall z \, (z \in X \leftrightarrow z \in Y) \rightarrow X = Y). A set and a class having the same extension are identical. Hence MK is not a two-sorted theory, appearances to the contrary notwithstanding.
Foundation Foundation may refer to: * Foundation (nonprofit), a type of charitable organization ** Foundation (United States law), a type of charitable organization in the U.S. ** Private foundation, a charitable organization that, while serving a good cause ...
: Each nonempty class ''A'' is disjoint from at least one of its members. :\forall A \not = \varnothing \rightarrow \exists b (b \in A \land \forall c (c \in b \rightarrow c \not\in A)) Class Comprehension: Let φ(''x'') be any formula in the language of MK in which ''x'' is a
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
and ''Y'' is not free. φ(''x'') may contain parameters that are either sets or proper classes. More consequentially, the quantified variables in φ(''x'') may range over all classes and not just over all sets; ''this is the only way MK differs from NBG''. Then there exists a class Y=\ whose members are exactly those sets ''x'' such that \phi(x) comes out true. Formally, if ''Y'' is not free in φ: :\forall W_1 ... W_n \exists Y \forall x \in Y \leftrightarrow (\phi(x, W_1, ... W_n) \land Mx)
Pairing In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''-modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-modu ...
: For any sets ''x'' and ''y'', there exists a set z=\ whose members are exactly ''x'' and ''y''. :\forall x \, \forall y \, (Mx_\land_My)_\rightarrow_\exists_z_\,_(Mz_\land_\forall_s_\,_[_s_\in_z_\leftrightarrow_(s_=_x_\,_\lor_\,_s_=_y).html" ;"title="s \in z \leftrightarrow (s = x \, \lor \, s = y)">(Mx \land My) \rightarrow \exists z \, (Mz \land \forall s \, [ s \in z \leftrightarrow (s = x \, \lor \, s = y)">s \in z \leftrightarrow (s = x \, \lor \, s = y)">(Mx \land My) \rightarrow \exists z \, (Mz \land \forall s \, [ s \in z \leftrightarrow (s = x \, \lor \, s = y) Pairing licenses the unordered pair in terms of which the
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
, \langle x,y \rangle, may be defined in the usual way, as \ \. With ordered pairs in hand, Class Comprehension enables defining relations and function (set theory), functions on sets as sets of ordered pairs, making possible the next axiom: Axiom of limitation of size, Limitation of Size: ''C'' is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
if and only if '' V'' can be mapped one-to-one into ''C''. :\begin \forall C lnot MC \leftrightarrow \exists F ( \forall x [Mx \rightarrow \exists s (s \in C \land \langle x, s \rangle \in F)\land \\ \qquad \forall x \forall y \forall s [(\langle x, s \rangle \in F \land \langle y, s \rangle \in F) \rightarrow x = y])]. \end The formal version of this axiom resembles the
axiom schema of replacement In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
, and embodies the class function ''F''. The next section explains how Limitation of Size is stronger than the usual forms of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
Power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
: Let ''p'' be a class whose members are all possible
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of the set ''a''. Then ''p'' is a set. :\forall a \, \forall p \, Ma_\land_\forall_x_\,_[x_\in_p_\leftrightarrow_\forall_y_\,_(y_\in_x_\rightarrow_y_\in_a)_\rightarrow_Mp.html" ;"title=" \in p \leftrightarrow \forall y \, (y \in x \rightarrow y \in a)">Ma \land \forall x \,
_\in_p_\leftrightarrow_\forall_y_\,_(y_\in_x_\rightarrow_y_\in_a)">Ma_\land_\forall_x_\,_[x_\in_p_\leftrightarrow_\forall_y_\,_(y_\in_x_\rightarrow_y_\in_a)_\rightarrow_Mp
_\in_p_\leftrightarrow_\forall_y_\,_(y_\in_x_\rightarrow_y_\in_a)">Ma_\land_\forall_x_\,_[x_\in_p_\leftrightarrow_\forall_y_\,_(y_\in_x_\rightarrow_y_\in_a)_\rightarrow_Mp Axiom_of_union">Union_ Union_commonly_refers_to: *_Trade_union,_an_organization_of_workers_ *_Union_(set_theory),_in_mathematics,_a_fundamental_operation_on_sets Union_may_also_refer_to: _Arts_and_entertainment _Music__ *_Union_(band),_an_American_rock_group_ **__''Un_...
:_Let_s=\bigcup_a_be_the_sum_class_of_the_set_''a'',_namely_the_union_(set_theory).html" "title="Axiom_of_union.html" "title=" \in p \leftrightarrow \forall y \, (y \in x \rightarrow y \in a) \rightarrow Mp"> \in p \leftrightarrow \forall y \, (y \in x \rightarrow y \in a)">Ma \land \forall x \, [x \in p \leftrightarrow \forall y \, (y \in x \rightarrow y \in a) \rightarrow Mp Axiom of union">Union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
: Let s=\bigcup a be the sum class of the set ''a'', namely the union (set theory)">union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of all members of ''a''. Then ''s'' is a set. :\forall a \, \forall s \, [(Ma \land \forall x \, [x \in s \leftrightarrow \exists y \, (x \in y \land y \in a)]) \rightarrow Ms]. axiom of infinity, Infinity: There exists an inductive set ''y'', meaning that (i) the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is a member of ''y''; (ii) if ''x'' is a member of ''y'', then so is x \cup \.. :\exists y y \land \varnothing \in y \land \forall z(z \in y \rightarrow \exists x [x \in y \land \forall w (w \in x \leftrightarrow [w = z \lor w \in z] )]. Note that ''p'' and ''s'' in Power Set and Union are universally, not existentially, quantified, as Class Comprehension suffices to establish the existence of ''p'' and ''s''. Power Set and Union only serve to establish that ''p'' and ''s'' cannot be proper classes. The above axioms are shared with other set theories as follows: * ZFC and NBG: Pairing, Power Set, Union, Infinity; * NBG (and ZFC, if quantified variables were restricted to sets): Extensionality, Foundation; * NBG: Limitation of Size.


Discussion

Monk (1980) and Rubin (1967) are set theory texts built around MK; Rubin's
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
includes
urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
s. These authors and Mendelson (1997: 287) submit that MK does what is expected of a set theory while being less cumbersome than ZFC and NBG. MK is strictly stronger than ZFC and its
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
NBG, the other well-known set theory with
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
es. In fact, NBG—and hence ZFC—can be proved consistent in MK. MK's strength stems from its axiom schema of Class Comprehension being
impredicative In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...
, meaning that φ(''x'') may contain quantified variables ranging over classes. The quantified variables in NBG's axiom schema of Class Comprehension are restricted to sets; hence Class Comprehension in NBG must be predicative. (Separation with respect to sets is still impredicative in NBG, because the quantifiers in φ(''x'') may range over all sets.) The NBG axiom schema of Class Comprehension can be replaced with finitely many of its instances; this is not possible in MK. MK is consistent relative to ZFC augmented by an axiom asserting the existence of strongly
inaccessible cardinal In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
s. The only advantage of the
axiom of limitation of size In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earli ...
is that it implies the
axiom of global choice In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an ele ...
. Limitation of Size does not appear in Rubin (1967), Monk (1980), or Mendelson (1997). Instead, these authors invoke a usual form of the local
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, and an "axiom of replacement," asserting that if the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
of a class function is a set, its
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
is also a set. Replacement can prove everything that Limitation of Size proves, except prove some form of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. Limitation of Size plus ''I'' being a set (hence the universe is nonempty) renders provable the sethood of the empty set; hence no need for an
axiom of empty set In axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstra ...
. Such an axiom could be added, of course, and minor perturbations of the above axioms would necessitate this addition. The set ''I'' is not identified with the
limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
\omega, as ''I'' could be a set larger than \omega. In this case, the existence of \omega would follow from either form of Limitation of Size. The class of
von Neumann ordinal In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s can be
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-orde ...
ed. It cannot be a set (under pain of paradox); hence that class is a proper class, and all proper classes have the same size as ''V''. Hence ''V'' too can be well-ordered. MK can be confused with second-order ZFC, ZFC with
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies onl ...
(representing second-order objects in set rather than predicate language) as its background logic. The language of second-order ZFC is similar to that of MK (although a set and a class having the same extension can no longer be identified), and their syntactical resources for practical proof are almost identical (and are identical if MK includes the strong form of Limitation of Size). But the
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (f ...
of second-order ZFC are quite different from those of MK. For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.


Model theory

ZFC, NBG, and MK each have models describable in terms of ''V'', the von Neumann universe of sets in ZFC. Let the
inaccessible cardinal In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
κ be a member of ''V''. Also let Def(''X'') denote the Δ0 definable
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of ''X'' (see
constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
). Then: * ''V''κ is model of ZFC; * Def(''V''κ) is a model of Mendelson's version of NBG, which excludes global choice, replacing limitation of size by replacement and ordinary choice; * ''V''κ+1, the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''V''κ, is a model of MK.


History

MK was first set out in and popularized in an appendix to
J. L. Kelley John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology, general topology and functional analysis. Kelley's 195 ...
's (1955) ''General Topology'', using the axioms given in the next section. The system of Anthony Morse's (1965) ''A Theory of Sets'' is equivalent to Kelley's, but formulated in an idiosyncratic formal language rather than, as is done here, in standard
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. The first set theory to include
impredicative In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...
class comprehension was Quine's ML, that built on
New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundations ...
rather than on ZFC.
Impredicative In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...
class comprehension was also proposed in Mostowski (1951) and
Lewis Lewis may refer to: Names * Lewis (given name), including a list of people with the given name * Lewis (surname), including a list of people with the surname Music * Lewis (musician), Canadian singer * "Lewis (Mistreated)", a song by Radiohead ...
(1991).


The axioms in Kelley's ''General Topology''

The axioms and definitions in this section are, but for a few inessential details, taken from the Appendix to Kelley (1955). The explanatory remarks below are not his. The Appendix states 181 theorems and definitions, and warrants careful reading as an abbreviated exposition of axiomatic set theory by a working mathematician of the first rank. Kelley introduced his axioms gradually, as needed to develop the topics listed after each instance of ''Develop'' below. Notations appearing below and now well-known are not defined. Peculiarities of Kelley's notation include: * He did ''not'' distinguish variables ranging over classes from those ranging over sets; * ''domain f'' and ''range f'' denote the domain and range of the function ''f''; this peculiarity has been carefully respected below; * His primitive logical language includes class abstracts of the form \ \, "the class of all sets ''x'' satisfying ''A''(''x'')." Definition: ''x'' is a ''set'' (and hence not a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
) if, for some ''y'', x \in y. I. Extent: For each ''x'' and each ''y'', ''x=y'' if and only if for each ''z'', z \in x when and only when z \in y. Identical to ''Extensionality'' above. I would be identical to the
axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...
in ZFC, except that the scope of I includes proper classes as well as sets. II. Classification (schema): An axiom results if in : For each \beta, \beta \in \ if and only if \beta is a set and B, 'α' and 'β' are replaced by variables, ' ''A'' ' by a formula Æ, and ' ''B'' ' by the formula obtained from Æ by replacing each occurrence of the variable that replaced α by the variable that replaced β provided that the variable that replaced β does not appear bound in ''A''. ''Develop'': Boolean algebra of sets. Existence of the null class and of the universal class ''V''. III. Subsets: If ''x'' is a set, there exists a set ''y'' such that for each ''z'', if z \subseteq x, then z \in y. The import of III is that of ''Power Set'' above. Sketch of the proof of Power Set from III: for any ''class'' ''z'' that is a subclass of the set ''x'', the class ''z'' is a member of the set ''y'' whose existence III asserts. Hence ''z'' is a set. ''Develop'': ''V'' is not a set. Existence of
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
s. Separation provable. IV. Union: If ''x'' and ''y'' are both sets, then x \cup y is a set. The import of IV is that of ''Pairing'' above. Sketch of the proof of Pairing from IV: the singleton \ of a set ''x'' is a set because it is a subclass of the power set of ''x'' (by two applications of III). Then IV implies that \ is a set if ''x'' and ''y'' are sets. ''Develop'': Unordered and
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s, relations,
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s,
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
,
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
,
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. V. Substitution: If ''f'' is a
lass Lass may refer to: *A girl/young woman in Scottish/Northern English People Surname *August Lass (1903–1962), Estonian footballer * Barbara Kwiatkowska-Lass (1940–1995), Polish actress *Donna Lass (1944–' 1970), possible victim of the Zodiac ...
function and ''domain f'' is a set, then ''range f'' is a set. The import of V is that of the
axiom schema of replacement In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
in NBG and ZFC. VI. Amalgamation: If ''x'' is a set, then \bigcup x is a set. The import of VI is that of ''Union'' above. IV and VI may be combined into one axiom.Kelley (1955), p. 261, fn †. ''Develop'':
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
,
injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...
,
surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
,
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
,
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
. VII. Regularity: If x \neq \varnothing there is a member ''y'' of ''x'' such that x \cap y = \varnothing. The import of VII is that of ''Foundation'' above. ''Develop'':
Ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s,
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...
. VIII. Infinity: There exists a set ''y'', such that \varnothing \in y and x \cup \ \in y whenever x \in y. This axiom, or equivalents thereto, are included in ZFC and NBG. VIII asserts the unconditional existence of two sets, the
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
inductive set ''y'', and the null set \varnothing. \varnothing is a set simply because it is a member of ''y''. Up to this point, everything that has been proved to exist is a class, and Kelley's discussion of sets was entirely hypothetical. ''Develop'':
Natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, N is a set,
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. Definition: ''c'' is a ''choice function'' if ''c'' is a function and c(x) \in x for each member ''x'' of ''domain c''. IX. Choice: There exists a choice function ''c'' whose domain is V - \.. IX is very similar to the
axiom of global choice In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an ele ...
derivable from ''Limitation of Size'' above. ''Develop'': Equivalents of the axiom of choice. As is the case with ZFC, the development of the
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s requires some form of choice. If the scope of all quantified variables in the above axioms is restricted to sets, all axioms except III and the schema IV are ZFC axioms. IV is provable in ZFC. Hence the Kelley treatment of MK makes very clear that all that distinguishes MK from ZFC are variables ranging over
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
es as well as sets, and the Classification schema.


Notes


References

* John L. Kelley 1975 (1955) ''General Topology''. Springer. Earlier ed., Van Nostrand. Appendix, "Elementary Set Theory." * Lemmon, E. J. (1986) ''Introduction to Axiomatic Set Theory''. Routledge & Kegan Paul. *
David K. Lewis David Kellogg Lewis (September 28, 1941 – October 14, 2001) was an American philosopher who is widely regarded as one of the most important philosophers of the 20th century. Lewis taught briefly at UCLA and then at Princeton University fr ...
(1991) ''Parts of Classes''. Oxford: Basil Blackwell. * The definitive treatment of the closely related set theory NBG, followed by a page on MK. Harder than Monk or Rubin. * Monk, J. Donald (1980) ''Introduction to Set Theory''. Krieger. Easier and less thorough than Rubin. * Morse, A. P., (1965) ''A Theory of Sets''. Academic Press. * . * Rubin, Jean E. (1967) ''Set Theory for the Mathematician''. San Francisco: Holden Day. More thorough than Monk; the ontology includes
urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
s. * .


External links


Download ''General Topology'' (1955) by John L. Kelley in various formats. The appendix contains Kelley's axiomatic development of MK.
From Foundations of Mathematics (FOM) discussion group:



{{DEFAULTSORT:Morse-Kelly set theory Systems of set theory